Promoting Mathematical Thinking
Noticing
John Mason
Stockholm
May 2013
The Open University
Maths Dept
1
University of Oxford
Dept of Education
It is only after you come to know
the surface of things
that you venture to see what is underneath;
But the surface of things is inexhaustible
(Italo Calvino)
He who loves practice without theory
is like the sailor
who boards ship without a rudder and compass
and never knows where he may cast.
Practice always rests on good theory.
(Leonardo Da Vinci)
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Say What You See
3
Say What You See
4
Say What You See
5
Say What You See
6
Say What You See
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Say What You See
Compare the areas
covered by the three
colours
Max Bill (artist)
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Gasket Sequences
é Wn+1
ê
êë Bn+1
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ù é 8 0 ù é Wn ù
ú=ê
ú
úê
úû ë 1 1 û êë Bn úû
é Wn+1
ê
êë Bn+1
ù é 8 1 ù é Wn ù
ú=ê
ú
úê
úû ë 1 8 û êë Bn úû
Say What You See
y=
(x – b)(x – c)(x – d)
(x – c)(x – d)(x – a)
A +
B
(a – b)(a – c)(a – d)
(b – c)(b – d)(b – a)
(x – d)(x – a)(x – b)
(x – a)(x – b)(x – c)
C +
D
+
(c – d)(c – a)(c – b)
(d – a)(d – b)(d – c)
(a, A)
(c, C)
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(b, B)
(d, D)
Pebble Arithmetic
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Observations About Observations

A name or label immediately shapes what is seen
and what is recalled
Frederick Bartlett 1932

Observation is theory laden
George Hanson 1958

We want our theories to be as fact
laden as our facts are theory laden
Nelson Goodman 1978
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Reporting Data
Accounting For
Account of
“They couldn’t …”
“They didn’t display
evidence of …”
“They can’t”
“They don’t display
evidence of …”
“I didn’t detect
evidence of …”
1
13
“I don’t detect
evidence of …”
Conjectures
Events consist of the stories told
The more precisely the data is specified
and
the more precise the analysis …
… the more we learn about the researcher’s
sensitivities to notice
and access to discourse to describe
1
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Protases



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I cannot change others;
I can work at changing myself
To express is to over stress
One thing we do not seem to learn from
experience …
… is that we do not often learn from
experience alone
Natural Epistemology
Noticing – Marking – Recording
 Conjecturing
 Resonance seeking

…with own experience
…with others
Avoid the teaching of
speculators,
whose judgements are
not confirmed
by experience.
(Leonardo Da Vinci)

Validity found in
…use by local community of practice
…own future practices informed
1
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Systematic Reflection
Keeping
Accounts
Seeking
Threads
Recognising Choices
Distinguishing
Choices
Accumulating
Alternatives
Preparing & Noticing
Imagining
Possibilities
Noticing
Possibilities
Identifying & labelling
Validating with Others
Describing
Moments
1
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Refining
Exercises
Specting
Intraspective
Interspective
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I
n
t
r
o
s
p
e
c
ti
v
e
Essence of Discipline of Noticing

Systematic Reflection
…Past (accounts-of not accounting-for)

Preparing & Noticing
…For Future & Present

Recognising Choices
…Could-have & Could-be
(not should have or should be)

Validating
…for Self & with Others
1
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Interwoven Worlds
Own
world of experience
Colleague's
world of experience
Reflecting
Trying
Seeking
resonance
with others
Recognising
Possibilities
Preparing
Expressing
World of observations
& theories
2
20
The universe is a mirror
in which we can contemplate
only
what we have learned
to know about ourselves
(Italo Calvino)
2
21
Adam can get irritable
and moody at times,
so I secretly replaced
his caffeinated coffee
with decaff
Let’s see if he notices
2
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I guess technically
to notice something
you have to be awake
Follow Up
Mason, J. (2002). Researching Your own Practice: The Discipline of Noticing,
London: RoutledgeFalmer.
Selected Articles
Mason, J. (2011). Noticing: roots and branches. In M. Sherin & R. Philips.
Mathematics Teacher Noticing: Seeing Through Teachers’ Eyes. p.35-50.
Mahwah: Erlbaum.
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